Where points came from

Abstract

The central thesis of this work, argued through references to authors widely separated in time (Pasch, Cantor and Hilbert, but also Plato and Euclid), is that the notion of point as an independent object inevitably produces paradoxes that can only be eliminated if we take points not as entities with an independent, primary reality of their own, but simply as reifications of a methodological choice, namely that of neglecting in pure theory any measuring mistake, however small. Thus, for example, we associate exact positions to a line’s extremes, with no indeterminacy at all, and call these positions points.